The answer to this question seems to be yes, because you can simulate it with a classical computer and thus by a local classical theory (rule 110 CA) (see this question). However most people disagree with this fact (that is classical), and I a would like to understand why. Why, for instance, is always the main argument against emergent quantum mechanics, such as the famous experiments with oil droplets on a vibrating fluid?
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No, entanglement is not a classical phenomenon. That is because its very definition is quantum:
Let $H_1, H_2$ be quantum spaces of states of two system. A quantum state $\chi\in H_1\otimes H_2$ of the combined system is called separable if it is a simple tensor, i.e. if there are $\phi\in H_1,\psi\in H_2$ such that $\chi = \phi\otimes\psi$. A state is called entangled if it is not separable.
And that's it. Classically, you cannot mimick this because the combined space of classical configuration spaces $Q_1,Q_2$ is the product $Q_1\times Q_2$, where by definition there are $q_1,q_2$ such that $q = (q_1,q_2)$ for every $q\in Q_1\times Q_2$.
In words, this is saying the following: To every classical state of a system there are uniquely specified states of the subsystems. For a quantum system, the non-entangled states are precisely the states where there are uniquely specified states of the subsystem from which they arise.
It has to be noted that not every entangled state shows "unclassical" behaviour in the sense that you cannot "see" (in the sense of, "model with classical theories") the same kind of correlation or result in some classical system. Entanglement is, in and of itself, a statement about the theory of states of systems, not about correlations and observations.